Integrand size = 15, antiderivative size = 27 \[ \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{x}-E(\arcsin (x)|-1)+\operatorname {EllipticF}(\arcsin (x),-1) \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {331, 313, 227, 1195, 435} \[ \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx=\operatorname {EllipticF}(\arcsin (x),-1)-E(\arcsin (x)|-1)-\frac {\sqrt {1-x^4}}{x} \]
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Rule 227
Rule 313
Rule 331
Rule 435
Rule 1195
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-x^4}}{x}-\int \frac {x^2}{\sqrt {1-x^4}} \, dx \\ & = -\frac {\sqrt {1-x^4}}{x}+\int \frac {1}{\sqrt {1-x^4}} \, dx-\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx \\ & = -\frac {\sqrt {1-x^4}}{x}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx \\ & = -\frac {\sqrt {1-x^4}}{x}-E\left (\left .\sin ^{-1}(x)\right |-1\right )+F\left (\left .\sin ^{-1}(x)\right |-1\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},x^4\right )}{x} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56
method | result | size |
meijerg | \(-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};x^{4}\right )}{x}\) | \(15\) |
default | \(-\frac {\sqrt {-x^{4}+1}}{x}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (F\left (x , i\right )-E\left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\) | \(53\) |
elliptic | \(-\frac {\sqrt {-x^{4}+1}}{x}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (F\left (x , i\right )-E\left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\) | \(53\) |
risch | \(\frac {x^{4}-1}{x \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (F\left (x , i\right )-E\left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\) | \(57\) |
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none
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx=-\frac {x E(\arcsin \left (x\right )\,|\,-1) - x F(\arcsin \left (x\right )\,|\,-1) + \sqrt {-x^{4} + 1}}{x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
Time = 0.39 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx=\frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]
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\[ \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 1} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 1} x^{2}} \,d x } \]
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Time = 5.69 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx=-\frac {{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ x^4\right )}{x} \]
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